Peter Ruzicka Sirocco 2004 0 Results and

Peter Ruzicka Sirocco 2004 0

Results and research directions in ATM and optical networks Shmuel Zaks Technion, Israel zaks@cs. technion. ac. il www. cs. technion. ac. il/~zaks Sirocco 2004 1

Sirocco 2004 2

Sirocco 2004 3

Sirocco 2004 4

References Works with O. Gerstel T. Eilam M. Shalom M. Feigelstein I. Cidon S. Moran M. Flammini Works of C. Kaklamanis E. Kranakis D. Krizanc A. Pelc I. Vrt’o V. Stacho G. Gambossi L. Bechetti D. Peleg J. C. Bermond A. Rosenberg L. Gargano and many more Sirocco 2004 5

• • graph-theoretic models algorithmic issues greedy constructions recursive constructions complexity issues approximation algorithms dynamic and fault-tolerance • combinatorial design issues • upper and lower bounds analysis • … • many open problems Sirocco 2004 6

Outline 4 4 ATM networks model Optical networks model Discussion – ATM networks Discussion – Optical networks Sirocco 2004 7

ATM Asynchronous Transfer Mode Transmission and multiplexing technique Industry standard for high-speed networks graph theoretic model Gerstel, Cidon, Zaks Sirocco 2004 8

Communication Virtual path Virtual channel concatenation of complete paths concatenation of partial paths Sirocco 2004 9

Cost Virtual path Virtual channel load = 3 (space) hop count = 2 (time) stretch factor = 4/3 Sirocco 2004 Other parameters 10

Find a layout, to connect a given node with all others, with given bounds on the load and the hop count Example: Sirocco 2004 11

Sirocco 2004 12

Outline 4 4 ATM networks model Discussion – ATM networks Optical networks model Discussion – Optical networks Sirocco 2004 13

Problem 1: Given a network, pairs of nodes and bounds h and l, find a virtual path layout to connect these nodes with the load bounded by l and the hop count bounded by h. Sirocco 2004 14

Sirocco 2004 15

Given a network and a bound on the load l and a bound h on the hop count, find a layout, to connect a given node with all others (one-to-all). a. worst-case. b. average case. Note: consider it for a given stretch factor. Problem 1 a: Sirocco 2004 16

Given a network and a bound on the load l and a bound h on the hop count, find a layout, to connect every two nodes (all-to-all). a. worst-case. b. average case. Note: consider it for a given stretch factor. Problem 1 b: Sirocco 2004 17

Problem 2: Input: Graph G, integers h, l > 0 , and a vertex v. Question: is there a VP layout for G, by which v can reach all other nodes, with hop count bounded by h and load bounded by l ? Sirocco 2004 18

load hop 1 2 3 . . 1 P P P … 2 P NP NP … 3 NP … … … . . . … … Flammini, Eilam, Zaks Sirocco 2004 19

Problem 1: Given a network, pairs of nodes and bounds h and l, find a virtual path layout to connect these nodes with the load bounded by l and the hop count bounded by h. tree, mesh general directed path network Gertsel, Wool, Zaks Feighelstein, Zaks Sirocco 2004 20

Case 1: shortest paths (stretch factor = 1) T(l, h) T(l-1, h) T(l, h-1) Sirocco 2004 21

Sirocco 2004 22

Sirocco 2004 23

Use of binary trees Sirocco 2004 24

Sirocco 2004 25

Sirocco 2004 26

Sirocco 2004 27

Case 2: any paths (stretch factor > 1) TL(l, h) TL(l-1, h) TR(l-1, h-1) Sirocco 2004 TL(l, h-1) 28

T(l, h-1) T(l-1, h) T(l-1, h-1) T(l, h-1) Sirocco 2004 29

l=3, h=2 Sirocco 2004 30

Golomb Sirocco 2004 31

Use of ternary trees Sirocco 2004 32

Using spheres The l 1 -norm |v| of an l-dimensional vector v = (x 1 , . . . , xl ) is defined as |v| = |x 1| + |x 2| +. . . + |xl| ex: |(1, -3, 0, 2)| = |1|+|-3|+|0|+|2| = 6 Sirocco 2004 33

Sp(l, r) - The l-dimensional l 1 -Sphere of radius h : the set of lattice points v=(x 1, . . . , xl) with distance at most h from the origin. Sp(2, 3): 2 - dimensional l 1 -Sphere of radius 3. point with l 1 -distance 3 from the origin. Sirocco 2004 34

Covering Radius The l - dimensional Covering Radius of N is the radius of the smallest ldimensional sphere containing at least N points |Sp(2, 0)| = 1 |Sp(2, 1)| = 5 |Sp(2, 2)| = 13 |Sp(2, 3)| = 25 Sirocco 2004 35

For every ATM Chain Layouts with N nodes and maximal load l: minimal radius of a layout with load l and N nodes minimal radius of an l-dimensional sphere with at least N internal points Sirocco 2004 36

load = 3 hop = 4 (1, 0, 0) dimension 3 radius = 4 (1, -1, 0) (1, -2, 0) (1, -3, 0) (0, 0, 0) (0, -1, 0) (-1, 0, 0) (-1, 1) (-1, 0) (-2, 0, 0) Sirocco 2004 37

the tree T(l, h) fills the sphere Sp(l, h) !!! |T(l, h)| = |T(h, l)| , hence |Sp(l, h)| = |Sp(h, l)| Sirocco 2004 38

Sp(2, 1): 2 - dimensional l 1 -Sphere of radius 1. Sp(1, 2): 1 - dimensional l 1 -Sphere of radius 2. Sirocco 2004 39

Using volume formulas, to Achieve bounds on h, given N and l For Upper Bound Sirocco 2004 40

Problem: Given a chain network with N nodes and a given bound on the maximum load, find an optimal layout with minimum hop count (or diameter ) between all pairs of nodes. Bounds for in: Kranakis, Krizanc, Pelc Stacho, Vrt’o Aiello, Bhatt, Chung, Rosenberg, Sitaraman Sirocco 2004 41

For every graph G with diameter D(G) and radius R(G): R(G) D(G) 2 R(G) Then: Sirocco 2004 42

one-to-all, all-to-all, some-to-some Design and analyze approximation algorithms for general network. Problem 3: Solve these problems to other measures (like load on the vertices, or bounded stretch factor) Problem 4: Sirocco 2004 43

Problem 7: Extend the duality results. Problem 8: Extend the use of geometry. Sirocco 2004 44

More Problem and parameters n n n n n what are the input and the output? network: tree, mesh, general, directed cost measure average vs. worst case complexity approximation algorithms routing dynamic, distributed cost of anarchy? … Sirocco 2004 45

Outline 4 4 ATM networks model Optical networks model Discussion – ATM networks Discussion – Optical networks Sirocco 2004 46

1 st generation the fiber serves as a transmission medium Electronic switch Optic fiber Sirocco 2004 47

2 nd generation Optical switch Sirocco 2004 48

A virtual topology Sirocco 2004 49

2 nd generation Routing in the optical domain Two complementing technologies: - Wavelength Division Multiplexing (WDM): Transmission of data simultaneously at multiple wavelengths over same fiber - Optical switches: the output port is determined according to the input port and the wavelength Sirocco 2004 50

Find a coloring with smallest number of wavelengths for a given set of lightpaths Example: Sirocco 2004 51

Outline 4 4 ATM networks model Optical networks model Discussion – ATM networks Discussion – Optical networks Sirocco 2004 52

Problem 1 : minimize the number of wavelengths Sirocco 2004 53

Smallest no. of wavelengths: Sirocco 2004 2 54

Problem 1 : minimize the number of wavelengths Problem 1 a : Complexity Problem 1 b: Special networks, general Sirocco 2004 55

Problem 1 c : Given pairs to be connected, design a routing with minimal load, and then color it with minimal number of colors Problem 1 d : Given pairs to be connected, design a routing and a coloring with minimal number of colors. ……many references Sirocco 2004 56

Problem 2 : minimize the number of switches Sirocco 2004 57

no. of ADMs ADM Sirocco 2004 58

Recall: smallest no. of wavelengths: 2 8 ADMs Sirocco 2004 59

Smallest no. of ADMs: 7 3 wavelengths Sirocco 2004 60

Problem 2 : minimize the number of switches Problem 2 a : complexity Problem 2 b : approximation algorithms Problem 2 c : trees, special general networks, Problem 2 d : given pairs to connect, design a routing and a coloring with smallest number of ADMs. Sirocco 2004 61

Problem 2 b : approximation algorithms clearly: result: Sirocco 2004 62

Ring network Gerstel, Lin, Sasaki Calinescu, Wan Sirocco 2004 63

Ring network Shalom, Zaks Sirocco 2004 64

Gerstel, Lin, Sasaki 1. Number the nodes from 0 to n-1 (how? ) 2. Color all lightpaths passing through or starting at node 0. Sirocco 2004 65

3. Scan nodes from 1 to n-1. Color each lightpath starting at i: The colors of the lightpaths ending at i are used first, and only then other colors are used, from lowest numbered first. While color is not valid for a lightpath, try next color. Sirocco 2004 66

1 0 2 14 3 13 4 12 5 11 6 10 7 9 8 Sirocco 2004 67

Color not valid… 1 0 2 14 3 13 4 12 5 11 6 10 7 9 8 Sirocco 2004 68

Calinescu, Wan Use maximum matchings at each node. Sirocco 2004 69

Shalom, Zaks Combine ideas from Gerstel, Lin, Sasaki Calinescu, Wan together with preprocessing of removing cycles, which uses an approximation algorithm Hurkens, Schrijver to find all cycles up to a given size. Sirocco 2004 70

Analysis: Use of linear programming to show we show a set of constraints that, together with cannot be satisfied. Sirocco 2004 71

Problem 1 : minimize the number of wavelengths. Problem 2 : minimize the number of switches. Problem 3 : find trade-offs between the two measures of number of switches and number of colors. Sirocco 2004 72

Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles. Eilam, Moran, Zaks fast and simple protection mehanism Sirocco 2004 73

e d g c a b f cost = 7 Sirocco 2004 74

Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles. Problem 4 a: Characterize the networks topologies G, in which any simple path can be extended to a simple cycle. Sirocco 2004 75

Answer: iff - G is randomly Hamltonian ( = each DFS tree is a path) , or - G is a ring, a complete graph, or a complete balanced bipartite graph Korach, Ostfeld Chartrand, Kronk Sirocco 2004 76

Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles. Problem 4 b : Input: A Graph G, a set of lightpaths in G, a number k. Question : is there a ring partition of cost k ? Liu, Li, Wan, Frieder Sirocco 2004 77

Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles. Problem 4 c: Design and analyze an approximation algorithm. Sirocco 2004 78

A trivial heuristics: Given a set of lightpaths D, extend each lightpath to a cycle by adding one lightpath. cost = 2 n or: ( |D|=n ) cost opt + n Sirocco 2004 79

question: is there a heuristics for which cost = opt + n ( < 1 ) ? answer: no. Sirocco 2004 80

question: is there a heuristics for which cost opt + k n (k < 1 ) ? answer: yes. cost opt + 3/5 n Sirocco 2004 81

Problem 4 c: Design and analyze an approximation algorithm. We showed the measure of total number of switches, thus : What about the saving in alg vs the saving in opt in the number of switches? Problem 4 d : Note: Sirocco 2004 82

Problem 5 : find a routing with linear filters. One-band routers: DEMUX Received Forwarded Flammini, Navara Sirocco 2004 83

Problem 5 : find a routing with linear filters. Problem 5 a : Is it always possible to find a routing? Sirocco 2004 84

No: One-band routers are not universal: u 1 v 1 w 1 z 1 u 2 v 2 w 2 z 2 u 3 v 3 w 3 z 3 Sirocco 2004 r 85

Problem 5 : find a routing with linear filters. Problem 5 b : Define other routers and explor etheir capabilities. Sirocco 2004 86

Problem 6 : Find a uniform all-to-all routing in a ring, using a minimum number of ADMs. N=13 j i Units of flow Cost: 13+5+3=21 ADMs Sirocco 2004 87

2 1 1 3 5 4 1 2 N=13 1 2 1 Sirocco 2004 88

Problem 6 a : What can be said about simple polygons? about non-simple polygons? Shalom, Zaks Sirocco 2004 89

More Problem and parameters n n n n n what are the input and the output? cost measure, worst case vs. average case. coloring and routing Wavelength convertion networks: specific, general complexity approximation algorithms Dynamic cost of anarchy? … Sirocco 2004 90

Questions ? Sirocco 2004 91

Thank You Sirocco 2004 92